A360348 - OEIS

%I #10 Feb 13 2023 03:44:41

%S 1,9,100,1381,22771,435138,9442049,229265109,6160375990,181559237499,

%T 5826147967201,202295647539886,7559401377952659,302570522540568557,

%U 12917629672442043340,586047019821392518293,28159186576616423049683,1428679795354280280080736,76329278834398327082152543

%N a(n) = [y^n*x^n/n] log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) for n >= 1.

%C Related sequence: A002426(n) = [y^n*x^n/n] log( Sum_{m>=0} (1 + y + y^2)^m * x^m ) for n >= 1.

%H Paul D. Hanna, <a href="/A360348/b360348.txt">Table of n, a(n) for n = 1..300</a>

%F a(n) ~ BesselI(0, 2) * n^(n+1). - _Vaclav Kotesovec_, Feb 12 2023

%e L.g.f.: A(x) = x + 9*x^2/2 + 100*x^3/3 + 1381*x^4/4 + 22771*x^5/5 + 435138*x^6/6 + 9442049*x^7/7 + 229265109*x^8/8 + 6160375990*x^9/9 + ...

%e a(n) equals the coefficient of y^n*x^n/n in the logarithmic series:

%e log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) = (y^2 + y + 1)*x + (y^4 + 6*y^3 + 9*y^2 + 6*y + 1)*x^2/2 + (y^6 + 15*y^5 + 63*y^4 + 100*y^3 + 63*y^2 + 15*y + 1)*x^3/3 + (y^8 + 28*y^7 + 242*y^6 + 872*y^5 + 1381*y^4 + 872*y^3 + 242*y^2 + 28*y + 1)*x^4/4 + (y^10 + 45*y^9 + 665*y^8 + 4430*y^7 + 14545*y^6 + 22771*y^5 + 14545*y^4 + 4430*y^3 + 665*y^2 + 45*y + 1)*x^5/5 + (y^12 + 66*y^11 + 1491*y^10 + 16002*y^9 + 91293*y^8 + 281220*y^7 + 435138*y^6 + 281220*y^5 + 91293*y^4 + 16002*y^3 + 1491*y^2 + 66*y + 1)*x^6/6 + ...

%e Exponentiation yields the g.f. of A360349:

%e exp(A(x)) = 1 + x + 5*x^2 + 38*x^3 + 391*x^4 + 5077*x^5 + 79535*x^6 + 1458264*x^7 + 30621237*x^8 + ... + A360349(n)*x^n + ...

%o (PARI) {a(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (1 + m*y + y^2)^m *x^m ) +x*O(x^n) ), n, x), n, y)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A360349, A360238, A002426.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 12 2023